CMSC 250 Homework 6 Fall 2001

Due Wed Oct 10 at the beginning of your discussion section.

This is due the day before exam 1 - this material will be on the exam. The answers will be posted after lab that evening. You must write the solutions to the problems single-sided on your own lined paper, with all sheets stapled together, and with all answers written in sequential order or you will lose points.

1. For each of the following statements, either prove that the statement is true, or give a counterexample to show that the statement is false.
   1. If $0 < r < 1$ is a real number, then $(\frac{1}{r})^{2}>\frac{1}{r}$.

   2. The product of a rational number with an irrational number is a rational number.

2. For each of the following statements, either prove that the statement is true, or give a counterexample to show that the statement is false.
   1. Let $a,b,c\in Z^{+}$. If $a \vert b$ and $b \vert c$, then $a \vert c$.

   2. Let $a, b\in{\bf {Z}}^{+}$. If $a \not{\vert} 8$ and $a \vert 8b$, then $a \vert b$.

3. Suppose that a computer program has been running for $num\_seconds$ seconds. Use div and mod notation to find the following numbers: $h$, $m$, and $s$ where $h$ can be any nonnegative integer, and $m$ and $s$ are both integers between 0 and 59 so that the values in the following output will be correct:

   The program has been running for $h$ hours, $m$ minutes, and $s$ seconds.

4. For each of the following statements, either prove that the statement is true, or give a counterexample to show that the statement is false.
   1. $\forall x\in{\bf {R}}$, $\lceil x^{3}\rceil = (\lceil x\rceil)^{3}.$

   2. $\forall x,y,z\in{\bf {R}}$, $\lfloor x +y+z\rfloor = \lfloor x\rfloor+\lfloor y\rfloor+\lfloor z\rfloor.$

5. Prove that the following statements are true.
   1. If $n + n^{2} + n^{3}$ is odd, then $n$ is odd.

   2. If $p$ is a prime and $a$ and $b$ are integers and $p \vert ab$ then $p \vert a$ or $p \vert b$.

   3. Prove that if $a\equiv b\hbox{ mod }n$ and $m \vert n$, then $a\equiv b \hbox{ mod } m$.

6. Suppose that $p$ and $q$ are prime numbers and that $p \vert a$ and $p \vert (a + q)$ for some integer $a$. Find all possible values for $p$.

7. Prove that the cube root of an irrational number is irrational.

8. Use the Unique Factorization theorem to prove that $\sqrt{5}$ is irrational.

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